# Existence of Quasilinear Neutral Impulsive Integrodifferential Equations in Banach Space

## Main Article Content

### Abstract

In this paper, we devoted to study the existence of mild solutions for quasilinear impulsive integrodi_erential equation in Banach spaces. The results are established by using Hausdor_'s measure of noncompactness and the _xed point theorems. Application is provided to illustrate the theory.

## Article Details

### References

- K. Balachandran and E. R. Anandhi, Neutral functional integrodifferential control systems in Banach spaces, Kybernetika, 39 (2003), 359-367.
- K. Balachandran and E. R. Anandhi, Controllability of neutral functional integrodifferential infinite delay systems in Banach spaces, Taiwanese Journal of Mathematics, 8 (2004), 689- 702.
- E. Hernandez and H. R. Henriquez, Impulsive partial neutral differential equations, Applied Mathematics Letters, 19 (2006), 215-222.
- J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, Springer-Verlag, New York, 1993.
- L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162 (1992), 494-505.
- L. Byszewski, H. Akca, Existence of solutions of a semilinear functional-differential evolution nonlocal problem, Nonlinear Anal. 34 (1998), 65-72.
- L. Byszewski, V. Lakshmikanthan, Theorems about the existence and uniqueness of solutions of a nonlocal Cauchy problem in Banach spaces, Applicable Anal. 40 (1990), 11-19.
- S. K. Ntouyas, P. Ch. Tsamatos, Global existence for semilinear evolution equations with nonlocal conditions, J. Math. Anal. Appl. 210 (1997), 679-687.
- Q. Dong, G. Li, J. Zhang, Quasilinear nonlocal integrodifferential equations in Banach spaces, Electronic J. Diff. Equa. 19 (2008), 1-8.
- A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
- D. Guo, X. Liu, External solutions of nonlinear impulsive integrodifferential equations in Banach spaces, J. Math. Anal. Appl. 177 (1993), 538-552.
- J. Liang, J. H. Liu, T. J. Xiao, Nonlocal impulsive problems for nonlinear differential equations in Banach spaces, Math. Computer Modelling. 49 (2009), 798-804.
- A. M. Samoilenko, N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.
- Z. Luo, J. J. Nieto, New results for the periodic boundary value problem for impulsive integrodifferential equations, Nonlinear Anal. 70 (2009), 2248-2260.
- J. Banas, K. Goebel, Measure of noncompactness in Banach spaces, Lecture Notes in Pure and Applied Mathematics, vol. 60, Marcle Dekker, New York, 1980.
- F.S. De Blasi, On a property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. R.S. Roumanie, 21 (1977), 259-262.
- J. M. Ayerbe, T. D. Benavides, G. L. Acedo, Measure of noncmpactness in in metric fixed point theorem, Birkhauser, Basel, 1997.
- J. Banas, W.G. El-Sayed, Measures of noncompactness and solvability of an integral equation in the class of functions of locally bounded variation, J. Math. Anal. Appl. 167 (1992), 133- 151.
- G. Emmanuele, Measures of weak noncompactness and fixed point theorems, Bull. Math. Soc. Sci. Math. R. S. Roumanie, 25 (1981), 353-358.
- Q. Dong, G. Li, The Existence of solutions for semilinear differential equations with nonlocal conditions in Banach spaces, Electronic J. Qualitative Theory of Diff. Equa. 47 (2009), 1-13.
- Z. Fan, Q. Dong, G. Li, Semilinear differential equations with nonlocal conditions in Banach spaces, International J. Nonlinear Sci. 2 (2006), 131-139.
- H.P. Heinz, On the behavior of measure of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal. 7 (1983), 1351-1371.
- M. Kamenskii, V. Obukhovskii, P. Zecca, Condensing multivalued maps and semilinear differential inclusions in Banach spaces, De Gruyter Series. Nonlinear Anal. Appl. Vol.7, de Gruyter, Berlin, 2001.
- D. Both, Multivalued perturbation of m-accretive differential inclusions, Israel J. Math. 108 (1998), 109- 138.
- K. Balachandran, J. Y. Park, Existence of solutions and controllability of nonlinear integrodifferential systems in Banach spaces, Mathematical Problems in Engineering, 2 (2003), 65-79.
- B. Radhakrishnan, K. Balachandran, Controllability results for semilinear impulsive integrodifferential evolution systems with nonlocal conditions, J. Control Theory and Appl. 10 (2012), 28-34.
- L. Wang and Z. Wang, Controllability of abstract neutral functional differential systems with infinite delay, Dynamics of Continuous, Discrete and Impulsive Systems Ser.B: Applications and Algorithms, 9 (2002), 59-70.